Quantitative Fourier decay for Patterson-Sullivan measures of dimension larger than $1/2$

Abstract

We give an elementary proof of power Fourier decay for Patterson-Sullivan measures associated to convex co-compact Schottky groups of dimension $\delta >\frac{1}{2}$, obtaining the explicit decay exponent $\frac{\delta (2\delta -1)}{(2\delta +1)(3-\delta )}$. The proof replaces the technical machinery used in previous works, such as sum-product estimates, renewal theory, Dolgopyat methods, and $L^2$ flattening, by elementary oscillatory integral estimates, hyperbolic geometry, and a duality argument based on the transpose Schottky group.

19 pages